Taregeted Maximum Likelihood Estimation for a Binary Outcome: Tutorial and Guided Implementation

1 Introduction

During the last 30 years, the modern epidemiology has been able to identify significant limitations of classic epidemiologic methods when the focus is to explain the main effect of a risk factor on a disease or outcome.

Causal Inference based on the Neyma-Rubin Potential Outcomes Framework (Rubin, 2011), first introduced in Social Science by Donal Rubin (Rubin, 1974) and later in Epidemiology and Biostatistics by James Robins (Greenland and Robins, 1986), has provided the theory and statistical methods needed to overcome recurrent problems in observational epidemiologic research, such as:

1. non-collapsibility of the odds and hazard ratios,
2. impact of paradoxical effects due to conditioning on colliders,
3. selection bias related to the vague understanding of the effect of time on exposure and outcome and,
4. effect of time-dependent confounding and mediators,
5. etc.

Classical epidemiologic methods to control for confounding require making the assumption that the effect measure is constant across levels of confounders included in the model.

Alternatively, James Robins in 1986 demonstrated that using standardisation, implemented through the use of the G-formula, allows obtaining a unconfounded marginal estimation of the causal average treatment effect (ATE) under causal untestable assumptions (Greenland and Robins, 1986), (Robins et al., 2000). The most commonly used estimator for a binary treatment effect is the risk difference or ATE = \(\psi(P_{0})\).

2 The G-Formula

\[\psi(P_{0})\,=\,\sum_{w}\,\left[\sum_{y}\,P(Y=y\mid A=1,W=w)-\,\sum_{y}\,P(Y = y\mid A=0,W=w)\right]P(W=w)\]

where,

\[P(Y = y \mid A = a, W = w)\,=\,\frac{P(W = w, A = a, Y = y)}{\sum_{y}\,P(W = w, A = a, Y = y)}\]
is the conditional probability distribution of Y = y, given A = a, W = w and,

\[P(W = w)\,=\,\sum_{y,a}\,P(W = w, A = a, Y = y)\]

• Classical epidemiologic methods require making the assumption that the effect measure is constant across levels of confounders included in the model. However, Standardisation allows obtaining an unconfounded summary effect measure without requiring this assumption. The G-formula is a generalisation of standardisation (Greenland and Robins, 1986).

• The ATE can be estimated non-parametrically using the G-formula. However, the course of dimensionality in observational studies limits its estimation.

• Hence, the estimation of the ATE using the G-formula relies mostly on parametric modelling assumptions and maximum likelihood estimation. The correct model specification in parametric modelling is crucial to obtain unbiased estimates of the true ATE (Rubin, 2011).

However, Mark van der Laan and collaborators have developed a double-robust estimation procedure to reduce bias against misspecification. The targeted maximum likelihood estimation (TMLE) is a semiparametric, efficient substitution estimator (Laan and Rose, 2011).

3 TMLE

TMLE allows for data-adaptive estimation while obtaining valid statistical inference based on the targeted minimum loss-based estimation and machine learning algorithms to minimise the risk of model misspecification (Laan and Rose, 2011). The main characteristics of TMLE are:

1. TMLE is a general algorithm for the construction of double-robust, semiparametric, efficient substitution estimators. TMLE allows for data-adaptive estimation while obtaining valid statistical inference.

2. TMLE implementation uses the G-computation estimand (G-formula). Briefly, the TMLE algorithm uses information in the estimated exposure mechanism P(A|W) to update the initial estimator of the conditional expectation of the outcome given the treatment and the set of covariates W, E\(_{0}\)(Y|A,W).

3. The targeted estimates are then substituted into the parameter mapping \(\Psi\). The updating step achieves a targeted bias reduction for the parameter of interest \(\psi(P_{0})\) (the true target parameter) and serves to solve the efficient score equation, namely the Influence Curve (IC). As a result, TMLE is a double robust estimator.

4. TMLE it will be consistent for \(\psi(P_{0})\) if either the conditional expectation E\(_{0}\)(Y|A,W) or the exposure mechanism P\(_{0}\)(A|W) are estimated consistently.

5. TMLE will be efficient if the previous two functions are consistently estimated achieving the lowest asymptotic variance among a large class of estimators. These asymptotic properties typically translate into lower bias and variance in finite samples (Bühlmann et al., 2016).

6. The general formula to estimate the ATE using the TMLE method:

\[\psi TMLE,n = \Psi(Q_{n}^{*})= {\frac{1}{n}\sum_{i=1}^{n}\bar{Q}_{n}^{1}\left(1,\ W_{i}\right)-\bar{Q}_{n}^{1}\left(0,\ W_{i}\right)}. (1)\] 7. The efficient influcence curve (IC) based on Hampel seminal paper (Hampel, 1974) is applied for statistical inference using TMLE:

\[IC_{n}(O_{i})\ \ =\ \left(\frac{I\left(A_{i}=1\right)}{g_n\left(1\left|W_{i}\right)\right)}\ -\ \frac{I\left(A_{i}=0\right)}{g_n\left(0\left|W_{i}\right)\right)}\ \right)\left[Y_{i}-\bar{Q}_{n}^{1}\left(A_{i},W_{i}\right)\right]+\bar{Q}_{n}^{1}\left(1,\ W_{i}\right)-\bar{Q}_{n}^{1}\left(0,\ W_{i}\right) - \psi TMLE,n. (2)\] where the variance of the ATE:

\[\sigma({\psi_{0}})=\sqrt{\frac{Var(IC_{n})}{n}}. (3)\]

8. The procedure is available with standard software such as the tmle package in R (Gruber and Laan, 2011).

4 Structural causal framework

4.1 Direct Acyclic Graph (DAG)

Figure 1. Direct Acyclic Graph
Source: Miguel Angel Luque-Fernandez

4.2 DAG interpretation

The ATE is interpreted as the population risk difference in one-year mortality for lung cancer patients diagnosed via emergency presentations versus non-emergency presentations. Under causal assumptions, and compared with non-emergency presentations of lung cancer, the risk difference of one-year mortality for emregency presentations increases by approximately 20%.

5 Causal assumptions

Under the counterfactual framework the following assumptions have to be considered to estimate the \(\psi(P_{0})\) (ATE) with a mondel for \(P_{0}\) augmented with additional nontestatble cuasal assumptions (Rubin, 2011), (Laan and Rose, 2011):

5.1 CMI or Randomization

(\(Y_{0},Y_{1}\perp\)A|W) of the binary treatment effect (A) on the outcome (Y) given the set of observed covariates (W), where W = (W1, W2, W3, … , Wk).

5.2 Positivity

a ϵ A: P(A=a | W) > 0
P(A=1|W=w) > 0 and P(A=0| W = w) > 0 for each possible w.

5.3 Consistency or SUTVA:

The observed outcome value, under the observed treatment, is equal to the counterfactual outcome corresponding to the observed treatment for identical independent distributed (i.i.d.) variables.

6 TMLE flow chart

Figure 2. TMLE flow chart (Road map)
Source: Mark van der Laan and Sherri Rose. Targeted learning: causal inference for observational and experimental dataSpringer Series in Statistics, 2011.

7 Data generation

7.1 Simulation

In R we create a function to generate the data. The function will have as input number of draws and as output the generated observed data (ObsData) including the counterfactuals (Y1, Y0).

The simulated data replicationg the DAG in Figure 1:

1. Y: mortality binary indicator (1 death, 0 alive)
2. A: binary treatment for emergency presentation at cancer diagnosis (1 EP, 0 NonEP)
   
3. W1: Gender (1 male; 0 female)
   
4. W2: Age at diagnosis (0 <65; 1 >=65)
   
5. W3: Cancer TNM classification (scale from 1 to 4)
   
6. W4: Comorbidities (scale from 1 to 5)

#install.packages("broom")
options(digits=4)
generateData <- function(n){
  w1 <- rbinom(n, size=1, prob=0.5)
  w2 <- rbinom(n, size=1, prob=0.65)
  w3 <- round(runif(n, min=0, max=4), digits=3)
  w4 <- round(runif(n, min=0, max=5), digits=3)
  A  <- rbinom(n, size=1, prob= plogis(-0.4 + 0.2*w2 + 0.15*w3 + 0.2*w4 + 0.15*w2*w4))
  Y  <- rbinom(n, size=1, prob= plogis(-1 + A -0.1*w1 + 0.3*w2 + 0.25*w3 + 0.2*w4 + 0.15*w2*w4))
  
  # counterfactual
  Y.1 <- rbinom(n, size=1, prob= plogis(-1 + 1 -0.1*w1 + 0.3*w2 + 0.25*w3 + 0.2*w4 + 0.15*w2*w4))
  Y.0 <- rbinom(n, size=1, prob= plogis(-1 + 0 -0.1*w1 + 0.3*w2 + 0.25*w3 + 0.2*w4 + 0.15*w2*w4))
  
  # return data.frame
  data.frame(w1, w2, w3, w4, A, Y, Y.1, Y.0)
}
set.seed(7777)
ObsData <- generateData(n=10000)
True_Psi <- mean(ObsData$Y.1-ObsData$Y.0);
cat(" True_Psi:", True_Psi)

 True_Psi: 0.198

Bias_Psi <- lm(data=ObsData, Y~ A)
cat("\n")

cat("\n Naive_Biased_Psi:",summary(Bias_Psi)$coef[2, 1])

 Naive_Biased_Psi: 0.2631

Naive_Bias <- ((summary(Bias_Psi)$coef[2, 1])-True_Psi); cat("\n Naives bias:", Naive_Bias)

 Naives bias: 0.06509

Naive_Relative_Bias <- (((summary(Bias_Psi)$coef[2, 1])-True_Psi)/True_Psi)*100; cat("\n Relative Naives bias:", Naive_Relative_Bias,"%")

 Relative Naives bias: 32.88 %

7.2 Data visualization

# DT table = interactive
# install.packages("DT") # install DT first
library(DT)
datatable(head(ObsData, n = nrow(ObsData)), options = list(pageLength = 5, digits = 2))

8 TMLE simple implementation

8.1 Step 1: \(Q_{0}\)(A,W)

Estimation of the initial probability of the outcome (Y) given the treatment (A) and the set of covariates (W), denoted as \(Q_{0}\)(A,W). To estimate \(Q_{0}\)(A,W) we can use a standard logistic regression model:

\[\text{logit}[P(Y=1|A,W)]\,=\,\beta_{0}\,+\,\beta_{1}A\,+\,\hat{\beta_{2}^{T}}W.\]

Therefore, we can estimate the initial probability as follows:

\[\bar{Q}^{0}(A,W)\,=\,\text{expit}(\hat{\beta_{0}}\,+\,\hat{\beta_{1}}A\,+\,\hat{\beta_{2}^{T}}W).\]

The predicted probability can be estimated using the Super-Learner library implemented in the R package “Super-Learner” (Van der Laan et al., 2007) to include any terms that are functions of A or W (e.g., polynomial terms of A and W, as well as the interaction terms of A and W, can be considered).

Consequently, for each subject, the predicted probabilities for both potential outcomes \(\bar{Q}^{0}(0,W)\) and \(\bar{Q}^{0}(1,W)\) can be estimated by setting A = 0 and A = 1 for everyone respectively: \[\bar{Q}^{0}(0,W)\,=\,\text{expit}(\hat{\beta_{0}}\,+\,\hat{\beta_{2}^{T}}W),\] and,
\[\bar{Q}^{0}(1,W)\,=\,\text{expit}(\hat{\beta_{0}}\,+\,\hat{\beta_{1}}A\,+\,\hat{\beta_{2}^{T}}W).\] Note: see appendix one for a short introduction to the Super-Learner and ensemble learning techniques.

ObsData <-subset(ObsData, select=c(w1,w2,w3,w4,A,Y))
Y  <- ObsData$Y
A  <- ObsData$A
w1 <- ObsData$w1
w2 <- ObsData$w2
w3 <- ObsData$w3
w4 <- ObsData$w4
m  <- glm(Y ~ A + w1 + w2 + w3 + w4, family=binomial, data=ObsData)
Q  <- cbind(QAW = predict(m),
            Q1W = predict(m, newdata=data.frame(A = 1, w1, w2, w3, w4)),
            Q0W = predict(m, newdata=data.frame(A = 0, w1, w2, w3, w4)))
Q0 <- as.data.frame(Q)
Y1<-Q0$Q1W 
Y0<-Q0$Q0W
#Inverse logit (probability scale)
psi <- (exp(Y1)/(1+exp(Y1)) - exp(Y0)/(1+exp(Y0)))
Psi <- mean(exp(Y1)/(1+exp(Y1)) - exp(Y0)/(1+exp(Y0))); cat("\n Q0:", Psi)

 Q0: 0.1992

df <- round(cbind(Q0,psi), digits= 3)

Visualizing the first step:

datatable(head(df, n = nrow(df)), options = list(pageLength = 5, digits = 3))

8.2 Step 2: \(g_{0}(A,W)\)

Estimation of the probability of the treatment (A) given the set of covariates (W), denoted as \(g_{0}(A,W)\). We can use again a logistic regression model and to improve the prediction algorithm we can use the Super-Learner library or any other machine learning strategy:

\[\text{logit}[P(A=1|W)]\,=\,\beta_{0}\,+\,\beta_{1}^{T}W.\] Then, we estimate the predicted probability of P(A|W) = \(\hat{g}(1,W)\) using:

\[\hat{g}(1,W)\,=\,\text{expit}\,(\hat{\beta_{0}}\,+\,\hat{\beta_{2}^{T}}W).\]

g <- glm(A ~ w2 + w3 + w4, family = binomial)
g1W = predict(g, type ="response");cat("\n Propensity score = g1W","\n");summary(g1W)

 Propensity score = g1W 
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  0.358   0.594   0.681   0.671   0.759   0.875 

8.3 Step 3: HAW and \(\epsilon\)

This step aims to find a better prediction model targeted at minimising the mean squared error (MSE) for the potential outcomes. For the ATE on step convergence is guaranteed given \(\bar{Q}^{0}\) and \(\hat{g}(1,W)\).

The fluctuation parameters \((\hat{\epsilon}_{0}\,,\,\hat{\epsilon}_{1})\) are estimated using maximum likelihood procedures by setting \(\text{logit}(\bar{Q^{0}}(A,W))\) as an offset in a intercept-free logistic regression with \(H_{0}\) and \(H_{1}\) as independent variables:

\[\bar{Q^{1}}(A,W)\,=\,\text{expit}\left[\text{logit}\left(\bar{Q^{0}}(A, W)\right)\,+\,\hat{\epsilon_{0}}H_{0}(A,W)\,+\,\hat{\epsilon_{1}}H_{1}(A,W)\right] (5)\] \[\bar{Q^{1}}(0,W)\,=\,\text{expit}\left[\text{logit}\left(\bar{Q^{0}}(A,W)\right)\,+\,\hat{\epsilon_{0}}H_{0}(0,W)\right]\]

\[\bar{Q^{1}}(1,W)\,=\,\text{expit}\left[\text{logit}\left(\bar{Q^{0}}(A,W)\right)\,+\,\hat{\epsilon_{1}}H_{1}(1,W)\right]\] Where, \[H_{0}(A,W)\,=\,\frac{I(A=0)}{\hat{g}(0|W)}\;\text{and},\;H_{1}(A,W)\,=\,\frac{I(A=1)}{\hat{g}(1|W)}\] are the stabilized inverse probability treatment (A) weights (IPTW), called the clever covariates and I defines an indicator function (note that \(\hat{g}(A|W)\) is estimted from step 2).

#Model 5: Clever covariate and fluctuating/substitution paramteres
h <- cbind(gAW=(A/g1w -(1-A)/(1-g1w)), g1W=(1/g1w), g0W=(-1/(1-g1w)))
epsilon <- coef(glm(Y ~ -1 + h[,1] + offset(Q[,"QAW"]), family = binomial));cat("\n Epsilon:",epsilon)

 Epsilon: 0.001189

df <- round(cbind(Q0,PS=(g1w),H=(h),epsilon), digits= 4)

Visualizing the 3rd step (PS = propensity score; H = IPTW or clever covarites):

datatable(head(df, n = nrow(df)), options = list(pageLength = 5, digits = 3))

8.4 Step 4 \(\bar{Q_{n}}^{*}:\text{from}\,\bar{Q_{0}}^{0}\,\text{to}\,\bar{Q_{1}}^{1}\)

Afterwards, the estimated probability of the potential outcomes is updated by the substitution parameters \((\hat{\epsilon_{0}}\,,\,\hat{\epsilon_{1}})\). The substitution update is performed by setting A = 0 and A = 1 for each subject in the initial estimate probability of the potential outcomes \(\bar{Q^{0}}(0,W)\,,\,\bar{Q^{0}}(1,W)\), as well as in the clever covariates \(H_{0}(0,W)\;\text{and}\; H_{1}(1,W)\).

For the \(\Psi(\bar{Q_{n}}^{*})\), the updated estimate of the potential outcomes only needs one iteration \(\Psi(\bar{Q_{n}}^{*})\) from \(\bar{Q}^{0}(A,W)\,=>\bar{Q^{1}}(A,W)\). Therefore, model (5) targets \(E[\hat{Y}_{A=0}]\;\text{and}\; E[\hat{Y}_{A=1}]\) simultaneously by including both \(H_{0}(A,W)\,\text{and}\,H_{1}(A,W)\) in the model. Hence \(\psi\) is finally estimated as follows:

\[\psi TMLE,n = \Psi(Q_{n}^{*})= {\frac{1}{n}\sum_{i=1}^{n}\bar{Q}_{n}^{1}\left(1,\ W_{i}\right)-\bar{Q}_{n}^{1}\left(0,\ W_{i}\right)}. (1)\]

Qstar <- plogis(Q + epsilon*h)
psi <- (Qstar[,"Q1W"] - Qstar[,"Q0W"])
Psi <- mean(Qstar[,"Q1W"] - Qstar[,"Q0W"]);
cat("TMLE_Psi:", Psi)

TMLE_Psi: 0.2004

cat("\n TMLE.SI_bias:", abs(True_Psi-Psi))

 TMLE.SI_bias: 0.002383

cat("\n Relative_TMLE.SI_bias:",abs(True_Psi-Psi)/True_Psi*100,"%")

 Relative_TMLE.SI_bias: 1.204 %

Visualizing the 4th step (H = IPTW or clever covarites):

df <- round(cbind(Q0=(Q0),H=(h),epsilon,psi), digits= 4)
datatable(head(df, n = nrow(df)), options = list(pageLength = 5, digits = 3))

cat("\n Psi first row:", plogis((0.001189*1.2399)+(2.3678))-(plogis((0.001189*-5.1681)+(1.3413))))

 Psi first row: 0.1228

cat("\n TMLE_Psi:", Psi)

 TMLE_Psi: 0.2004

8.5 Step 5: Inference

TMLE uses the efficient influence curve (IC) for inference (i.e., to obtain standard errors for \(\psi\)).

\[IC_{n}(O_{i})\ \ =\ \left(\frac{I\left(A_{i}=1\right)}{g_n\left(1\left|W_{i}\right)\right)}\ -\ \frac{I\left(A_{i}=0\right)}{g_n\left(0\left|W_{i}\right)\right)}\ \right)\left[Y_{i}-\bar{Q}_{n}^{1}\left(A_{i},W_{i}\right)\right]+\bar{Q}_{n}^{1}\left(1,\ W_{i}\right)-\bar{Q}_{n}^{1}\left(0,\ W_{i}\right) - \psi TMLE,n. (2)\]

where the standard deviation for \(\psi\) is estimated as follows:

\[\sigma({\psi_{0}})=\sqrt{\frac{Var(IC_{n})}{n}}. (3)\] Note: see appendix two for a short introduction to the theory of the Influence Curve.

Q <- as.data.frame(Q)
IC <- h[,1]*(Y-Q$QAW) + Q$Q1W - Q$Q0W - Psi;summary(IC)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 -2.950  -0.855  -0.200  -0.006   0.570  14.300 

n <- nrow(ObsData)
varHat.IC <- var(IC)/n; varHat.IC

[1] 0.0002024

#Psi and 95%CI for Psi
cat("\n TMLE_Psi:", Psi)

 TMLE_Psi: 0.2004

cat("\n 95%CI:", c(Psi-1.96*sqrt(varHat.IC),  Psi+1.96*sqrt(varHat.IC)))

 95%CI: 0.1725 0.2283

cat("\n TMLE.SI_bias:", abs(True_Psi-Psi))

 TMLE.SI_bias: 0.002383

cat("\n Relative_TMLE.SI_bias:",abs(True_Psi-Psi)/True_Psi*100,"%")

 Relative_TMLE.SI_bias: 1.204 %

9 TMLE vs. AIPTW

1. The advantages of TMLE have repeatedly been demonstrated in both simulation studies and applied analyses (Laan and Rose, 2011).

2. Evidence shows that TMLE provides the less unbiased ATE estimate compared with other double-robust estimators (Neugebauer and Laan, 2005), (Laan and Rose, 2011) such as the combination of regression adjustment with inverse probability of treatment weighting (IPTW-RA) and the augmented inverse probability of treatment weighting (AIPTW). The AIPTW estimation is a two-step procedure with two equations (propensity score equation and, mean outcome equation).

3. To estimate the ATE using the AIPTW estimator one can set the estimation equation (EE) (4) equal to zero and use bootstrap to derive 95% confidence intervals (CI). However, solving the EE using the generalized method of moments (GMM), stacking both equations (propensity score and outcome), reduces the estimation and inference steps to only one. However, given that the propensity score in equation (4) can easily fall outside the range [0, 1] (if for some observations \(g_{n}(1|W_{i})\) is close to 1 or 0) the AIPTW estimation can be unstable (near violation of the positivity assumption). AIPTW instability under near violation of the positivity assumption represents the price of not being a substitution estimator as TMLE.

\[\psi_{0}^{AIPTW-ATE}\ \ =\ \frac{1}{n}\sum_{i=1}^{n}\left(\frac{I\left(A_{i}=1\right)}{g_n\left(1\left|W_{i}\right)\right)}\ -\ \frac{I\left(A_{i}=0\right)}{g_n\left(0\left|W_{i}\right)\right)}\ \right)\left[Y_{i}-\bar{Q}_{n}^{0}\left(A_{i},W_{i}\right)\right]+\frac{1}{n}\sum_{i=1}^{n}\bar{Q}_{n}^{0}\left(1,\ W_{i}\right)-\bar{Q}_{n}^{0}\left(0,\ W_{i}\right). (4)\]

AIPTW <- mean((h[,1]*(Y-Q$QAW))+(Q$Q1W-Q$Q0W)); AIPTW

[1] 0.1948

cat("\n AIPTW_bias:", abs(True_Psi-AIPTW))

 AIPTW_bias: 0.003211

cat("\n Relative_AIPTW_bias:",abs(True_Psi-AIPTW)/True_Psi*100,"%")

 Relative_AIPTW_bias: 1.622 %

Compared with AIPTW, TMLE showed smaller relative bias.

10 TMLE using the Super-Learner

With TMLE we can call the Super-Learner (SL). The SL is a R-package using V-fold cross-validation and ensembled learning (prediction using the all the predictions of multiple stacked learning algorithms) techniques to improve model prediction performance (Breiman, 1996).

The basic implementation of TMLE in the R-package tmle uses by default three algorithms:
1. SL libraries SL.glm (main terms logistic regression of A and W),
2. SL.step (stepwise forward and backward model selection using AIC criterion, restricted to second order polynomials) and,
3. SL.glm.interaction (a glm variant that includes second order polynomials and two by two interactions of the main terms included in the model).

The principal interest of calling the Super-Learner is to obtain the less-unbiased estimated for \(\bar Q_{n}^{0}(A,W)\) and \(g_{0}(A,W)\). It is achieved by obtaining the smallest expected loss function for Y or A (binary outcomes), respectively. For instance, the negative logarithmic loss function for Y is computed as the minimizer of the expected squared error loss:
\[\bar Q_{0}\,=\, \text{arg min}_{\bar Q}E_{0}L(O, \bar Q),\]
where \(L(O, \bar Q)\) is: \[ (Y \,-\, \bar Q(A, W))^{2}\] Note: see appendix one for a short introduction to the Super-Learner and ensemble learning techniques.

1. Step One: \(\bar Q_{n}^{0}(A,W)\) prediction

#Q0
library(SuperLearner)
#Specify SuperLearner libraries
SL.library <- c("SL.glm","SL.step","SL.glm.interaction")
#Data frame with X with baseline covariates and exposure
X <- subset(ObsData, select=c(A, w1, w2, w3, w4))
n <- nrow(ObsData)
#Create data frames with A=1 and A=0
X1<-X0<-X
X1$A <-1
X0$A <-0
#Create new data by stacking
newdata <- rbind(X,X1,X0)
#Call superlearner
Qinit <- SuperLearner(Y=ObsData$Y, X=X, newX=newdata, SL.library=SL.library, family="binomial")
Qinit

Call:  SuperLearner(Y = ObsData$Y, X = X, newX = newdata, family = "binomial", SL.library = SL.library) 

                         Risk   Coef
SL.glm_All             0.1767 0.6024
SL.step_All            0.1767 0.0000
SL.glm.interaction_All 0.1768 0.3976

#Predictions
#Pred prob of survival given A, W
QbarAW <- Qinit$SL.predict[1:n]
#Pred prob of surv for each subject given A=1 and w
Qbar1W <- Qinit$SL.predict[(n+1):(2*n)]
#Pred prob of surv for each subject given A=0 and w
Qbar0W <- Qinit$SL.predict[(2*n+1):(3*n)]
#Simple substitution estimator Psi(Q0)
PsiHat.SS <- mean(Qbar1W-Qbar0W);PsiHat.SS

[1] 0.199

2. Step two: \(g_{0}(A,W)\) prediction

#Step 2 g_0(A|W) with SuperLearner
w <- subset(ObsData, select=c(w1,w2,w3,w4))
gHatSL <- SuperLearner(Y=ObsData$A, X=w, SL.library=SL.library, family = binomial)
gHatSL;mean(gHatSL)

Call:  SuperLearner(Y = ObsData$A, X = w, family = binomial, SL.library = SL.library) 

                         Risk   Coef
SL.glm_All             0.2092 0.0000
SL.step_All            0.2092 0.3641
SL.glm.interaction_All 0.2091 0.6359
[1] NA

#Generate the pred prob of A=1 and, A=0 given covariates
gHat1W <- gHatSL$SL.predict
gHat0W <- 1-gHat1W
#Step 3: Clever covariate
HAW <- as.numeric(ObsData$A==1)/gHat1W - as.numeric(ObsData$A==0)/gHat0W;mean(HAW)

[1] 0.00284

H1W <-  1/gHat1W
H0W <- -1/gHat0W

3. Steps 3 and 4: fluctuation step and substitution estimation for for for \(\bar Q_{n}^{0}(A,W)\) to \(\bar Q_{n}^{1}(A,W)\)

#Step 4: Substitution estimaiton Q* of the ATE.
logitUpdate <- glm(ObsData$Y ~ -1 + offset(qlogis(QbarAW))+HAW, family='binomial')
eps <- logitUpdate$coef;eps

      HAW 
0.0004375 

#Calculating the predicted values for each subject under each txt
QbarAW.star <- plogis(qlogis(QbarAW)+eps*HAW)
Qbar1W.star <- plogis(qlogis(Qbar1W)+eps*H1W)
Qbar0W.star <- plogis(qlogis(Qbar0W)+eps*H0W)
PsiHat.TMLE.SL <- mean(Qbar1W.star) - mean(Qbar0W.star)
cat("PsiHat.TMLE.SL:", PsiHat.TMLE.SL)

PsiHat.TMLE.SL: 0.1994

cat("\n PsiHat.TMLE.SL_bias:", abs(True_Psi-PsiHat.TMLE.SL))

 PsiHat.TMLE.SL_bias: 0.001447

cat("\n Relative_PsiHat.TMLE.SL_bias:",abs(True_Psi-PsiHat.TMLE.SL)/True_Psi*100,"%")

 Relative_PsiHat.TMLE.SL_bias: 0.7306 %

11 R-TMLE

Using the R-package tmle.

The basic implementation of TMLE in the R-package tmle uses by default three algorithms:
1. SL libraries SL.glm (main terms logistic regression of A and W),
2. SL.step (stepwise forward and backward model selection using AIC criterion, restricted to second order polynomials) and,
3. SL.glm.interaction (a glm variant that includes second order polynomials and two by two interactions of the main terms included in the model).

library(tmle)
w <- subset(ObsData, select=c(w1,w2,w3,w4))
tmle <- tmle(Y, A, W=w)
cat("TMLER_Psi:", tmle$estimates[[2]][[1]],";","95%CI(", tmle$estimates[[2]][[3]],")")

TMLER_Psi: 0.1994 ; 95%CI( 0.1799 0.2189 )

cat("\n TMLE_bias:", abs(True_Psi-tmle$estimates[[2]][[1]]))

 TMLE_bias: 0.001387

cat("\n Relative_TMLE_bias:",abs(True_Psi-tmle$estimates[[2]][[1]])/True_Psi*100,"%")

 Relative_TMLE_bias: 0.7006 %

12 R-TMLE improving prediction

In addition to the default algorithms implemented in the R-tmle package, we can improve our estimation calling more efficient machine learing algorithms, such as generalized additive models, the Random Forest and, Recursive Partitioning and Regression Trees:

SL.TMLER.Psi <- tmle(Y=Y, A=A, W=w, family="binomial", 
    Q.SL.library = c("SL.glm", "SL.step", "SL.glm.interaction", "SL.gam", "SL.randomForest", "SL.rpart"),
    g.SL.library = c("SL.glm", "SL.step", "SL.glm.interaction", "SL.gam", "SL.randomForest", "SL.rpart"))
cat("SL.TMLER.Psi:", SL.TMLER.Psi$estimates[[2]][[1]],";","95%CI(", SL.TMLER.Psi$estimates[[2]][[3]],")")

SL.TMLER.Psi: 0.1994 ; 95%CI( 0.1799 0.2188 )

cat("\n SL.TMLER.Psi_bias:", abs(True_Psi-SL.TMLER.Psi$estimates[[2]][[1]]))

 SL.TMLER.Psi_bias: 0.001357

cat("\n Relative_SL.TMLER.Psi_bias:",abs(True_Psi-SL.TMLER.Psi$estimates[[2]][[1]])/True_Psi*100,"%")

 Relative_SL.TMLER.Psi_bias: 0.6852 %

We have demonstrated:

1. TMLE excels the AIPTW estimator and,
   
2. TMLE best performance is obtained when calling more advanced Super-Learner algorithms.

13 Appendix One

Efron in 1982 showed that the empirical Influence Curve estimate of standar error is the same as the one obtained using the infinitesimal jackknife and the nonparametric deltha method (Efron and Efron, 1982).

1. The Delat Method uses the first order of the Taylor series expansion: \[f(x) \approx f(x)\,-\,f(\mu)\,+\,(x\,-\,\mu)f'(\mu);\]
   where \(f'(\mu)\) is the derivative of the function with respect to X evaluated at the mean of X. Therefore, squaring both terms, the variance is approximately estimate as follwos: \[\text{E}[f(x)-f(\mu)]^{2}\approx \text{E}(x\,-\,\mu)^{2}\,\times\,[f'(\mu)]^{2};\] The left-hand side of the above equation is approximately the variance of \(f(x)\) and applied to the empirical distribution of X, the sample estimate of variance for X replaces: \[\text{E}(x_{i}\,-\,\mu)^{2}.\] The infinitesimal jackknife estimate of the standard error is defined as follows: \[SD_{ij}(\theta_{e})\,=\,\left(\frac{\sum_{i=1}^{n}U_{i}^{2}}{n^{2}}\right)^{1/2};\] where \(\theta_{e}\) is the estimate of the parameter \(\theta\) and \(U_{i}\) is a directional derivative in the direction of the ith coordinate centered at the mean of the empirical distribution function.

Figure 3. Estimate of the \(\psi\) Standard Error using the efficient Influence Curve.
Image credit: Miguel Angel Luque-Fernandez.

14 Appendix Two

With TMLE we can call the R-package Super-Learner (SL). The SL uses cross-validation and ensembled learning (using all the predictions of multiple stacked learning algorithms) techniques to improve model prediction performance (Breiman, 1996).

The SL algorithm provides a system based on V-fold cross-validation (Efron and Gong, 1983) (10-folds) to combine adaptively multiple algorithms into an improved estimator, and returns a function than can be used for prediction in new datasets.

Figure 4: 10-fold cross-validation algorithm.
Source: Sherri Rose in Esembles in Public Health and Healt Policy. May 4, 2016.

The basic implementation of TMLE in the R-package tmle uses by default three algorithms: 1. SL libraries SL.glm (main terms logistic regression of A and W),
2. SL.step (stepwise forward and backward model selection using AIC criterion, restricted to second order polynomials) and, 3. SL.glm.interaction (a glm variant that includes second order polynomials and two by two interactions of the main terms included in the model).

The principal interest of calling the Super-Learner is to obtain the less-unbiased estimated for \(\bar Q_{n}^{0}(A,W)\) and \(g_{0}(A,W)\). It is achieved by obtaining the smallest expected loss function for Y or A (binary outcomes), respectively. For instance, the negative logarithmic loss function for Y is computed as the minimizer of the expected squared error loss:
\[\bar Q_{0}\,=\, \text{arg min}_{\bar Q}E_{0}L(O, \bar Q),\]
where \(L(O, \bar Q)\) is: \[ (Y \,-\, \bar Q(A, W))^{2}\] The SL algorithm first split the data into ten blocks and fits each of the selected algoriths on the training set (non-shaded blocks), then predicts the estimated probabilities of the outcome (Y) using the validation set (shaded block) for each algorithm, based on the corresponding training set. Afterwards, the SL estimates the the cross-validating risk for each algorithm averaging the risks across validation sets resulting in one estimated cross-validated risk for each algorithm. Finally, the SL selects the combination of Z that minimises the cross-validation risk, defined as the minimum mean square error for each of the selected algorithms using Y and Z. A weighted combination of the algorithms (ensemble learning) in Z is then used to predict the outcome (Y) (see Figure 5).

Figure 5: Flow Diagram for the Super-Learner algorithm.
Source: Mark van der Laan and Sherri Rose. Targeted learning: causal inference for observational and experimental dataSpringer Series in Statistics, 2011.

15 Session Info

devtools::session_info()

Session info --------------------------------------------------------------------------------------------------

 setting  value                       
 version  R version 3.3.0 (2016-05-03)
 system   x86_64, darwin13.4.0        
 ui       RStudio (1.0.44)            
 language (EN)                        
 collate  en_US.UTF-8                 
 tz       Europe/London               
 date     2016-11-01                  

Packages ------------------------------------------------------------------------------------------------------

 package      * version  date       source                            
 assertthat     0.1      2013-12-06 CRAN (R 3.3.0)                    
 base64enc      0.1-3    2015-07-28 CRAN (R 3.3.0)                    
 codetools      0.2-14   2015-07-15 CRAN (R 3.3.0)                    
 devtools       1.12.0   2016-06-24 CRAN (R 3.3.0)                    
 digest         0.6.10   2016-08-02 CRAN (R 3.3.0)                    
 DT           * 0.2      2016-08-09 CRAN (R 3.3.0)                    
 evaluate       0.10     2016-10-11 CRAN (R 3.3.0)                    
 foreach      * 1.4.3    2015-10-13 CRAN (R 3.3.0)                    
 formatR        1.4      2016-05-09 CRAN (R 3.3.0)                    
 gam          * 1.14     2016-09-10 CRAN (R 3.3.0)                    
 htmltools      0.3.5    2016-03-21 CRAN (R 3.3.0)                    
 htmlwidgets    0.7      2016-08-02 CRAN (R 3.3.0)                    
 iterators      1.0.8    2015-10-13 CRAN (R 3.3.0)                    
 jsonlite       1.1      2016-09-14 CRAN (R 3.3.0)                    
 knitr          1.14     2016-08-13 CRAN (R 3.3.0)                    
 magrittr       1.5      2014-11-22 CRAN (R 3.3.0)                    
 memoise        1.0.0    2016-01-29 CRAN (R 3.3.0)                    
 nnls         * 1.4      2012-03-19 CRAN (R 3.3.0)                    
 randomForest * 4.6-12   2015-10-07 CRAN (R 3.3.0)                    
 Rcpp           0.12.7   2016-09-05 CRAN (R 3.3.0)                    
 rmarkdown      1.1.9007 2016-10-25 Github (rstudio/rmarkdown@746d0eb)
 rpart        * 4.1-10   2015-06-29 CRAN (R 3.3.0)                    
 rprojroot      1.0-2    2016-03-28 CRAN (R 3.3.0)                    
 rsconnect      0.5      2016-10-17 CRAN (R 3.3.0)                    
 rstudioapi     0.6      2016-06-27 CRAN (R 3.3.0)                    
 stringi        1.1.2    2016-10-01 CRAN (R 3.3.0)                    
 stringr        1.1.0    2016-08-19 CRAN (R 3.3.0)                    
 SuperLearner * 2.0-19   2016-02-04 CRAN (R 3.3.0)                    
 tibble         1.2      2016-08-26 CRAN (R 3.3.0)                    
 tmle         * 1.2.0-4  2014-03-09 CRAN (R 3.3.0)                    
 withr          1.0.2    2016-06-20 CRAN (R 3.3.0)                    
 yaml           2.1.13   2014-06-12 CRAN (R 3.3.0)                    

16 Thank you

Thank you for participating in this tutorial.
If you have updates or changes that you would like to make, please send me a pull request. Alternatively, if you have any questions, please e-mail me. You can cite this repository as:
Luque-Fernandez MA, (2016). Taregeted Maximum Likelihood Estimation for a Binary Outcome: Tutorial and Guided Implementation. GitHub repository, http://migariane.github.io/TMLE.nb.html.
Miguel Angel Luque Fernandez
E-mail: miguel-angel.luque at lshtm.ac.uk
Twitter @WATZILEI

17 References

Breiman L. (1996). Stacked regressions. Machine learning 24: 49–64.

Bühlmann P, Drineas P, Laan M van der, Kane M. (2016). Handbook of big data. CRC Press.

Efron B, Efron B. (1982). The jackknife, the bootstrap and other resampling plans. SIAM.

Efron B, Gong G. (1983). A leisurely look at the bootstrap, the jackknife, and cross-validation. The American Statistician 37: 36–48.

Greenland S, Robins JM. (1986). Identifiability, exchangeability, and epidemiological confounding. International journal of epidemiology 15: 413–419.

Gruber S, Laan M van der. (2011). Tmle: An r package for targeted maximum likelihood estimation. UC Berkeley Division of Biostatistics Working Paper Series.

Hampel FR. (1974). The influence curve and its role in robust estimation. Journal of the American Statistical Association 69: 383–393.

Laan M van der, Rose S. (2011). Targeted learning: Causal inference for observational and experimental data. Springer Series in Statistics.

Neugebauer R, Laan M van der. (2005). Why prefer double robust estimators in causal inference? Journal of Statistical Planning and Inference 129: 405–426.

Robins JM, Hernan MA, Brumback B. (2000). Marginal structural models and causal inference in epidemiology. Epidemiology 550–560.

Rubin DB. (2011). Causal inference using potential outcomes. Journal of the American Statistical Association.

Rubin DB. (1974). Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of educational Psychology 66: 688.

Van der Laan MJ, Polley EC, Hubbard AE. (2007). Super learner. Statistical applications in genetics and molecular biology 6.